$A$ metal rod having a coefficient of linear expansion $2 \times 10^{-5} /{ }^{\circ} C$ is $0.75 \ m$ long at $45^{\circ} C$. When the temperature rises to $65^{\circ} C$,the increase in length of the rod will be: (in $mm$)

What must be the lengths of steel and copper rods at $0^o C$ for the difference in their lengths to be $10\,cm$ at any common temperature? $(\alpha_{steel}=1.2 \times 10^{-5} \;^o C^{-1})$ and $(\alpha_{copper} = 1.8 \times 10^{-5} \;^o C^{-1})$

$A$ unit scale is to be prepared whose length does not change with temperature and remains $20\,cm$,using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that the difference between their lengths remains constant. If the length of brass is $40\,cm$,what is the length of iron in $cm$?
($\alpha_{\text{iron}} = 1.2 \times 10^{-5} K^{-1}$ and $\alpha_{\text{brass}} = 1.8 \times 10^{-5} K^{-1}$)

$A$ metal rod of silver at $0^{\circ}C$ is heated to $100^{\circ}C$. Its length increases by $0.19\, cm$. If the original length of the rod is $100\, cm$,the coefficient of cubical expansion of the silver rod is:

$A$ steel tape measures the length of a copper rod as $90 \ cm$ when both are at $10 \ ^oC$. What length (in $cm$) would the tape read for the length of the rod when both are at $30 \ ^oC$? $[\alpha_{st} = 1.2 \times 10^{-5} / ^oC$ and $\alpha_{cu} = 1.7 \times 10^{-5} / ^oC]$

$A$ circular copper ring at $30^{\circ} C$ has a hole with an area of $9.98 \ cm^2$. It is made to slip onto a steel rod of cross-sectional area of $10 \ cm^2$, by raising the temperature of both the ring and the rod simultaneously by an amount $\Delta T$. If the coefficients of linear expansion of copper and steel are $17 \times 10^{-6} /{ }^{\circ} C$ and $11 \times 10^{-6} /{ }^{\circ} C$ respectively, then the minimum value of $\Delta T$ should be: (in $^{\circ} C$)